POPULATION DYNAMICS
Katja Goldring, Francesca Grogan, Garren Gaut,
Advisor: Cymra Haskell
Iterative Mapping

We iterate over a function starting at an initial x

Each iterate is a function of the previous
iterate

Two types of mappings
 Autonomous-
non-time dependent
 Non-autonomous- time dependent
Autonomous Systems

Chaotic System- doesn’t converge to a fixed
point given an initial x
A fixed point exists
wherever f(x) = x.
This serves as a tool for
visualizing iterations
fixed point
fixed point
Stability



Autonomous Systems

Autonomous Pielou Model:
a1 = 0.500000
2
1.8
1.8
1.6
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1.4
1.4
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1.2
a1 = 2.000000
2
1.4
1.6
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2
0
Carrying Capacity
carrying capacity
0
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1
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2
Autonomous Systems

Autonomous Sigmoid Beverton Holt:
a1 = 3.000000, delta1 = 2.000000
2.5
2
1.5
Allee Threshold
1
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0
0
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Graphs
a1 = 2.000000, delta1 = 2.000000
a1 = 3.000000, delta1 = 2.000000
2
2.5
1.8
1.6
2
1.4
1.2
1.5
1
0.8
1
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0
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Graphs
a1 = 0.500000, delta1 = 1.000000
a1 = 2.000000, delta1 = 1.000000
2
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
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Graphs
a1 = 1.300000, delta1 = 0.300000
a1 = 1.000000, delta1 = 2.000000
2
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
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1
1
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Non-Autonomous Systems

Pielou Logistic Model
Semigroup



A semigroup is closed and associative for an
operator
We want a set of functions to be a semigroup
under compositions
A fixed point of a composed function is an orbit
for a sequence of functions
Known Results

…Known Results
An Extended Model

Sigmoid Beverton-Holt Model
Known Results


Our Model

Sigmoid Beverton-Holt Model with varying
deltas and varying a’s.

Goal: We want to show that there exists a nontrivial stable periodic orbit for a sequence of
Sigmoid Beverton-Holt equations with varying
a’s and varying deltas.
Problems

We know a non-trivial periodic orbit doesn’t
exist for certain parameters, even in the
autonomous case

How to group functions for which we know an
orbit exists

Can we make a group of functions closed
under composition?
Lemmas


5
5
delta1
delta2
composition
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
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1.5
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1
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delta1
delta2
composition
4.5
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0
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Application to model
Corollary
The Stochastic Sigmoid Beverton Holt

We are now looking at the same model,
except we now pick our
each iteration.
and
randomly at
Density


A probability density function of a continuous
random variable is a function that describes the
likelihood of a variable occurring over a given
interval.
We are interested in how the density function on the
evolves. We conjecture that it will converge to
a unique invariant density. This means that after a
certain number of iterations, all initial densities will
begin to look like a unique invariant density.
Stochastic Iterative Process

We iterate over a function of the form
where the parameters
are chosen from independent distributions.
Stochastic Iterative Process

At each iterate, n, let
denote the density of
,, and let
denote the density of
 For
each iterate
is invariant, since we are always
picking our
from the same distribution.
 For each iterate
can vary, since where
falls varies on every iterate.

Since the distributions for
and
are
independent, the joint distribution of
and
is
Previous Results


Haskell and Sacker showed that for a Beverton-Holt
model with a randomly varying environment, given
by
there exists a unique invariant density to which all
other density distributions on the state variable
converge.
This problem deals with only one parameter and the
state variable.
…more Previous Results


Bezandry, Diagana, and Elaydi showed that the
Beverton Holt model with a randomly varying
survival rate, given by
has a unique invariant density.
Thus they were looking at two parameters, and the
state variable.
Stochastic Sigmoid Beverton Holt

We examine the Sigmoid Beverton Holt equation given
by
We’d like to show that under the restrictions
there exists a unique invariant density to which all other density
distributions on the state variable converge.
Our function
a1 = 1.300000, delta1 = 0.300000
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
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2
Our Method of Attack


We have
where
is a Markov Operator that acts on
densities.
We found an expression for
the stochastic
kernel of . , where
Method of Attack Continued


Lasota-Mackey Approach:
The choice of depends on what restrictions we
put on our parameters. We are currently refining
these.
Spatial Considerations


1-dimensional case where populations lie in a line
of boxes:
Goal: See if this new mapping still has a unique,
stable, nontrivial fixed point.
MATLAB visualizations
07/10/11
Implicit Function Theorem


We can use this to show existence of a fixed point
in F.
Application to Spatial Beverton-Holt
Banach Fixed Point Theorem


The previous theorem only guaranteed existence of
fixed point, whereas if we prove our map is a
contraction mapping, we can get uniqueness and
stability.
Application to Spatial Beverton-Holt


Application to Spatial Beverton-Holt


The conditions on the previous slide form halfplanes. We need to show the intersection of these
planes is invariant under F. We found this is the
case when
Therefore F is a contraction mapping on
when
the above conditions are satisfied, and F has
unique, stable fixed point on
.
Future Work

Finish the proof that the Sigmoid Beverton Holt
model has a unique invariant distribution under our
given restrictions.
 Expand
this result to include more of the Sigmoid
Beverton Holt equations.

Contraction Mapping: Prove there exists a q<1 such
that
References
Thanks to




Our advisor, Cymra Haskell.
Bob Sacker, USC.
REU Program, UCLA.
SEAS Café.